Calorimeter Shower Superresolution with Conditional Normalizing Flows: Implementation and Statistical Evaluation

Andrea Cosso

physics.ins-detMar 26, 2026

Calorimeter Shower Superresolution with Conditional Normalizing Flows: Implementation and Statistical Evaluation

Authors: Andrea Cosso

Organizations: Università degli Studi di Genova Scuola di Scienze Matematiche, Fisiche e Naturali Laurea Magistrale in Fisica

Abstract

In High Energy Physics, detailed calorimeter simulations and reconstructions are essential for accurate energy measurements and particle identification, but their high granularity makes them computationally expensive. Developing data-driven techniques capable of recovering fine-grained information from coarser readouts, a task known as calorimeter superresolution, offers a promising way to reduce both computational and hardware costs while preserving detector performance. This thesis investigates whether a generative model originally designed for fast simulation can be effectively applied to calorimeter superresolution. Specifically, the model proposed in arXiv:2308.11700 is re-implemented independently and trained on the CaloChallenge 2022 dataset based on the Geant4 Par04 calorimeter geometry. Finally, the model's performance is assessed through a rigorous statistical evaluation framework, following the methodology introduced in arXiv:2409.16336, to quantitatively test its ability to reproduce the reference distributions.

Figures & tables

Panel of Figure 1 — Figure 1: Projected CPU requirements. Left: Atlas [ 8 ] Right: CMS [ 18 ]

Figure 1.1 : Overfitting example. By fitting noisy data with sufficiently high degree polynomials, the curve is able to parametrize noise.

Figure 2.1 : Overview of a variational autoencoder (VAE) [ 55 ] . The encoder $q_{\phi}(\mathbf{z}|\mathbf{x})$ maps data $\mathbf{x}$ to latent variables $\mathbf{z}$ , while the decoder $p_{\theta}(\mathbf{x}|\mathbf{z})$ reconstructs the data from samples drawn from the latent space.

Panel of Figure 2 — Figure 2.2 : Schematic of a Generative Adversarial Network (GAN): the generator maps noise to data samples, which are evaluated by the discriminator alongside real data to predict fake or real . The diagram is strongly inspired by Ref. [ 82 ] .

Figure 2.3 : Diagram of the diffusion process. Image from Ref. [ 46 ]

Figure 2.4 : Illustration of density flow in a conditional flow-matching framework, adapted from [ 34 ] . The figure shows the continuous evolution of the probability density $p(z(t))$ governed by an ODE solver performing optimal transport between a simple Gaussian base distribution $p(z(t_{0}))$ and the complex target distribution $p(z(t_{1}))$ . The central panel depicts the vector field that drives the transformation, while the top and bottom panels show the marginal densities at the start and end of the flow.

Figure 2.5 : Illustration of the bidirectional mapping in Normalizing Flows In the generative direction (top), a simple latent variable sampled from a base distribution (typically a standard Gaussian) is transformed through a sequence of invertible mappings into a complex target distribution in data space. Conversely, in the normalizing direction (bottom), data samples are mapped back to the latent space, allowing for exact likelihood evaluation via the change-of-variables formula. The deformation of the background grid visually represents the smooth and invertible transformations that characterize flow-based models.

Figure 2.6 : On the left there is the conventional autoencoder with three hidden layers. The networks are oriented from top to bottom and the input passes through three fully connected layers. Note that since the output attempts at reconstructing the inputs, and it depends entirely on the inputs, the standard Autoencoder is not suitable for data generation. On the Right the networks has the same structure as the standard Autoencoder, but some connections have been removed, such that each input unit is only predicted based on the previous ones, using multiplicative binary masks. In the example, the ordering of the inputs has been changed to 3,1,2. Masks are constructed following the discussion illustrated in Section 2.3.1 .

Figure 2.7 : Example of a RQS transformation defined on the interval $\mathbb{B}=[-7,7]$ , with 6 bins and 7 knots.

Figure 2.8 : Illustration of Type I and Type II errors for a test statistic $t$ . The blue curve is the sampling distribution $f(t\,|\,H_{0})$ under the null hypothesis, and the orange curve is $f(t\,|\,H_{1})$ under the alternative. A decision threshold $t^{\star}$ defines the shaded $H_{0}$ rejection region (to the right). The Type I error rate $\alpha$ is the blue tail area beyond $t^{\star}$ (rejecting $H_{0}$ when $H_{0}$ is true). The Type II error rate $\beta$ is the orange area to the left of $t^{\star}$ (failing to reject $H_{0}$ when $H_{1}$ is true).

Figure 3.1 : Schematic cross-section of the ATLAS detector (not to scale). From the beam pipe outward: silicon pixel and strip trackers and the Transition Radiation Tracker inside a solenoid; the liquid-argon electromagnetic calorimeter; the TileCal hadronic calorimeter; and the outer muon spectrometer. Example particle signatures are shown: an electron and a photon producing electromagnetic showers, hadrons showering in the hadronic calorimeter, a muon traversing to the muon system, and a neutrino leaving no visible trace (seen as missing transverse momentum). Adapted from ATLAS public outreach material.

Panel of Figure 3 — Figure 3.2 : Energy loss in lead (right) and copper (left).

Figure 3.4 : Expected residual light yield $I/I_{0}$ at the end of HL-LHC (assumed $4000\penalty\ \mathrm{fb}^{-1}$ and a dose rate $7\times$ Run 2) for each TileCal barrel cell. Layer letters encode the radial depth ( A inner, B/BC middle, D outer); indices encode the $\eta$ -bin. Shaded ranges in [ 4 ] Table 2 summarize typical cells per layer (A: $48\text{\textendash}60\%$ loss; B/BC: $17\text{\textendash}25\%$ ; D: $7\text{\textendash}16\%$ ), while specific worst-case cells (e.g. A12/A13 with $66\text{\textendash}69\%$ loss) are called out explicitly. Figure adapted from [ 4 , Figure 18] .

Figure 3.5 : Average relative light yield $I/I_{0}$ for TileCal A13 cells versus simulated dose $d$ (bottom axis) and integrated luminosity $L$ (top axis). Points show in-situ measurements: 137 Cs (dots) and minimum-bias currents (triangles). Vertical lines mark end of Run 3 ( $\sim 530\penalty\ \mathrm{fb}^{-1}$ ) and end of HL-LHC ( $4000\penalty\ \mathrm{fb}^{-1}$ ). The solid curve represents the extrapolation including dose-rate effects expected at HL-LHC, while the dashed line, represents the extrapolation neglecting dose-rate. Image taken from [ 4 ] .

Figure 4.1 : Schematic view of the voxelization of the detector. In the specific example the detector has 3 layers, 3 radial bins, and 6 angular bins.

Figure 4.2 : A schematic for the detector of the Par04 example. It consists of layers of absorber (red) and active material (blue). The energy deposit is recorded in a cylindrical readout (black).

Figure 4.3 : 3D diagram from Ref. [ 62 ] illustrating the calorimeter voxel geometry for Dataset 2 (fine representation) in Grey and the two choices of voxellization introduced in Section 4.2.1 . In Red is shown the choice A and in Green the choice B .

Choice	Deposited energy	Incident energy	Fine layer energies	Neighbors energy	Coarse layer number	Coarse radial bin	Total
A	1	1	5	6	9	9	31
B	1	1	1	6	45	3	57

Table 4.1 : Number of each component of the conditional input for each choice of coarse voxelization.

Figure 4.4 : Effect of preprocessing on the raw distribution. In Blue the log-uniform distribution of the raw data and in Red , the distribution of the preprocessed data. Notice how the values of the preprocessed data are bound in the range $[-13.82,13.82]$ given by $\log(\alpha)$ and $\log(1/\alpha)$ , with $\alpha=1\cdot 10^{-6}$ .

Panel of Figure 4 — Figure 4.5 : Conditional MAF architecture with eight bijectors interleaved by alternating permutations.

Model	Base distribution dimensions	Number of MADE blocks	MADE layer sizes			RQS bins
Model	Base distribution dimensions	Number of MADE blocks	Input	Hidden	Output	RQS bins
A	10	8	41	$2\times 128$	230	8
B	12	8	69	$2\times 128$	276	8

Table 4.2: Configuration of the different flow models. The input layer dimension is defined to be the sum of the number of dimensions in the base distribution and the number of conditional inputs, discussed in Section 4.2.2 . Note that

2\times 128

is a shorthand notation for 2 hidden layers of 128 nodes each.

Figure 4.6 : Schematic of a conditional MADE block used inside a MAF. Three variables $\vec{x}=(x_{0},x_{1},x_{2})$ are transformed while four context variables $\vec{c}=(c_{0},\dots,c_{3})$ provide conditioning. Inputs are split into two parallel streams (one for $\vec{x}$ , one for $\vec{c}$ ), each fully connected to a width-6 hidden layer; their pre-activations are then added element-wise (the “ $+$ ” row), so conditioning enters without breaking the autoregressive masking. Two width-6 hidden layers are shown. The output layer produces three vectors $\vec{k}_{i}$ (one per transformed dimension $x_{i}$ ); each has 5 components implementing a two-bin Rational Quadratic Spline (one derivative and two knot coordinate pairs $(x_{1},y_{1})$ , $(x_{2},y_{2})$ ). Colored edges illustrate autoregressive dependencies: in red, $\vec{k}_{1}$ depends only on $x_{0}$ and $\vec{c}$ ; in blue, $\vec{k}_{2}$ depends on $x_{0},x_{1}$ and $\vec{c}$ . Choice B is only reported for completeness, in this thesis we only explore choice A .

Figure 4.7 : Three-phase One-Cycle learning rate schedule used for model training. We chose 30 epochs of warm-up, 46 of cooldown, and 4 epochs of annihilation.

Parameter	Value
Optimizer	Adam
LR scheduler	OneCycle
Initial learning rate	$1\times 10^{-5}$
Max learning rate	$8\times 10^{-4}$
$(\beta_{1},\beta_{2},\epsilon)$	(0.9, 0.999, $10^{-8}$ )
Batch size	10k

Table 4.3 : Summary of training hyperparameters.

Figure 4.9 : Plot of the validation and training losses over the course of training. Lower validation loss value was obtained at epoch 80.

Figure 4.10 : A comparison between runs with different batch size. There is a clear difference of training time if we compare a batch size of $10$ k or $30$ k. The absolute difference in total training time is about six hours.

Parameter	This work	Reference run (Ref. [ 62 ] )
Warm-up epochs	30	18
Cooldown epochs	46	18
Annihilation epochs	4	4
Batch size	10 000	60 000
Base learning rate ( $\alpha_{\text{base}}$ )	$1\times 10^{-5}$	$2\times 10^{-5}$
Maximum learning rate ( $\alpha_{\text{max}}$ )	$8\times 10^{-4}$	$1\times 10^{-3}$

Table 4.5: Comparison of training configurations between the present work and the reference implementation.

Figure 4.11 : Centroids distributions along the longitudinal, angular and radial direction. Truth samples ( Geant4 ) are filled in blue and generated samples are reported as orange line.

Figure 4.12 : RMS distributions along the longitudinal, lateral and (radial), and angular direction. Reference distribution is filled in blue, while results from generated showers is reported in orange.

Figure 4.13 : Distributions of the total energy deposited in layers 1, 10, 20 and 45. Can be obtained by summing the energy deposition in each fine voxel in radial and angular direction. The distribution of the Geant4 is filled in blue while the orange line represent the generated samples.

Feature	Metric	batch_size	n_slices	$p$ -value	Significance ( $\sigma$ )
Coarse shower	KS	1000	-	0.495	$1.3\cdot 10^{-2}$
	SKS		500	0.487	$3.3\cdot 10^{-2}$
	SWD		500	0.470	$7.6\cdot 10^{-2}$
Coarse shower	KS	5000	-	0.503	$8\cdot 10^{-3}$
	SKS		500	0.505	$1.3\cdot 10^{-2}$
	SWD		500	0.489	$2.8\cdot 10^{-2}$

Table 4.6 : Summary of

p

-values obtained for different configurations and metrics.

Figure 4.14 : Null vs. alternative hypothesis distributions of three test statistics at full dimensionality ( $d=6480$ ; batch_size $=1\cdot 10^{3}$ ; nslices $=1\cdot 10^{3}$ ). Each panel shows density-normalized histograms on a log $y$ -axis: the null (truth–truth, red fill) and the alternative (truth–model, blue outline). From left to right: average Kolmogorov–Smirnov (KS), Sliced KS (SKS), and Sliced Wasserstein distance (SWD). Vertical dashed lines mark the 68%, 95%, and 99% confidence levels estimated from the null; the shaded bands indicate the corresponding right-tailed rejection regions. A rightward shift of the blue curve relative to the red one signals increased power (greater separation from the null).

Figure 4.15 : Same conventions as Figure 4.14 . Here $d=6480$ , batch_size $=5\cdot 10^{3}$ and nslices $=1\cdot 10^{3}$ . Increased the batch size to probe the hyperparameter space.

Figure 4.16 : Same conventions as Figure 4.14 . Here $d=648$ , batch_size $=1\cdot 10^{3}$ and nslices $=5\cdot 10^{2}$ . The metrics are not able to reject $H_{0}$ .

Figure 4.17 : Same conventions as Figure 4.14 . Here $d=648$ , batch_size $=5\cdot 10^{3}$ and nslices $=5\cdot 10^{2}$ . The metrics are not able to reject $H_{0}$ .

Figure 4.18 : Same conventions as Figure 4.14 . Here $d=6480$ , batch_size $=1\cdot 10^{3}$ and nslices $=7\cdot 10^{3}$ to investigate the role of the slicing in the discrimination performance of the tests.

Feature	Dimensions	Metric	batch_size	n_slices	$p$ -value	Significance ( $\sigma$ )
Layer centroid	3	KS	1000	-	0.505	0.0125
		SKS		100.0	0.494	0.0150
		SWD		100.0	0.440	0.151
		MMD		-	0.498	0.00501
		FGD		-	0.476	0.0601
Layer centroid	3	KS	5000	-	0.462	0.0954

Table 4.7 : Summary of

p

-values from all the evaluation of physically inspired features.

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